No, they do not HAVE to wear them, but you will find most of them like wearing them, and they are also written into sponsorship contracts that players must wear them, for advertising purposes.
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as long as u can see the whole uniform, anything. on cold days if pitchers hit they usually are allowed to wear jackets. pitchers arent allowed to wear a whole lot on the field except there uniform and an undershirt. the undershirt is not allowed to be white as it is the same colour as the ball and can confuse the batter